Alan Turing proposed in 1950 a framework called an imitation game to decide
if a machine could think. Using mathematics developed largely after Turing --
category theory -- we analyze a broader class of universal imitation games
(UIGs), which includes static, dynamic, and evolutionary games. In static
games, the participants are in a steady state. In dynamic UIGs, "learner"
participants are trying to imitate "teacher" participants over the long run. In
evolutionary UIGs, the participants are competing against each other in an
evolutionary game, and participants can go extinct and be replaced by others
with higher fitness. We use the framework of category theory -- in particular,
two influential results by Yoneda -- to characterize each type of imitation
game. Universal properties in categories are defined by initial and final
objects. We characterize dynamic UIGs where participants are learning by
inductive inference as initial algebras over well-founded sets, and contrast
them with participants learning by conductive inference over the final
coalgebra of non-well-founded sets. We briefly discuss the extension of our
categorical framework for UIGs to imitation games on quantum computers.

图灵提出了 1950 年模仿游戏的框架，通过在范畴理论中使用数学，我们分析了更广泛的通用模仿游戏（UIGs），包括静态、动态和进化游戏，并使用两个有影响力的 Yoneda 结果来描述每种类型的模仿游戏。